Questions and Answers
If $(x + 3)^2 = 225$, which of the following could be the value of $x - 1?$ (Select all that apply)
For the equation $x^2 - 2x = 0$, which is bigger: Quantity A: $x$ or Quantity B: 2?
Can't say
Which is bigger: Quantity A: $d(d^2 - 2d + 1)$ or Quantity B: $d(d^2 - 2d) + 1$?
Can't say
With $a = 2b = 4c$, which quantity is bigger: Quantity A: $a + b$ or Quantity B: $a + c$?
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With $k = 2m = 4n$, which quantity is bigger: Quantity A: $km$ or Quantity B: $kn$?
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For the equations $3x + 6y = 27$ and $x + 2y + z = 11$, which is bigger: Quantity A: $z + 5$ or Quantity B: $x + 2y - 2$?
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With $a = b/2$ and $c = 3b$, which is bigger: Quantity A: $a$ or Quantity B: $c$?
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If it takes 20 students 1.5 hours to erect a bonfire, how long will it take 35 students to do the job? (Select one)
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For integers a, b, and c, if the sum of a and b is 75% of c, which is bigger: Quantity A: $(3/4)(a + b)$ or Quantity B: $(4/3)(c)$?
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For positive integers a, b, c, and d, if a is half of b and b is one-third of c, which is bigger: Quantity A: $(a + b)/c$ or Quantity B: $(a + b + c)/d$?
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If $a != b$, which is bigger: Quantity A: $(a-b)/(b-a)$ or Quantity B: 1?
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If $x > y$ and $x != y$, which is bigger: Quantity A: $x^2 / (x + 1/x)$ or Quantity B: $y^2 / (y + 1/y)$?
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If a is inversely proportional to b and a = 16 when b = 1, what is b when a = 8?
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If a is directly proportional to b, and a = 8 when b = 2, what is a when b = 4?
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If it takes 20 students 1.5 hours to erect a bonfire, how long will it take 35 students to do the job? (Select one)
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For integers a, b, and c, if the sum of a and b is 75% of c, which is bigger: Quantity A: $(3/4)(a + b)$ or Quantity B: $(4/3)(c)$?
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If $a + b = 8$, $b + c = 11$, and $a + c = 5$, what is the value of $a + b + c$?
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Study Notes
Card 1 - Quadratic Equation
- Solve (x + 3)² = 225 leads to x = 11 or x = -15.
- Potential values for x - 1 are examined; only x = 11 is valid, resulting in 12.
Card 2 - Quantity Comparison
- Equation x² - 2x = 0 yields x values of 0 and 2.
- Comparison between x and 2 is indeterminate since x can take two values.
Card 3 - Expression Evaluation
- Compare QA: d(d² - 2d) and QB: d(d² - 2d) + 1.
- Uncertainty exists due to unknown value of d.
Card 4 - Integer Relationships
- Given a = 2b = 4c, determine if a + b or a + c is greater.
- The result is uncertain without knowing the signs of a, b, and c.
Card 5 - Nonnegative Integer Comparison
- From k = 2m = 4n, evaluate quantities km vs. kn.
- Cannot determine which is larger as k, m, and n may be zero.
Card 6 - Linear Equations
- From the equations 3x + 6y = 27 and x + 2y + z = 11, both expressions z + 5 and x + 2y - 2 are equal.
- No need for individual computations reveals equality.
Card 7 - Proportional Relationships
- With a = b/2 and c = 3b, comparing a vs. c is inconclusive.
- Unknowns about the signs of a, b, or c hinder definitive assessment.
Card 8 - Inverse Proportionality
- Time to erect a bonfire is inversely proportional to the number of students.
- If 20 students take 1.5 hours, 35 students will take approximately 51 minutes.
Card 9 - Sum Relationships
- For integers a, b, and c where a + b = 75% of c, the comparison of expressions depends on potential negativity.
- Hence, the quantities can't be clearly evaluated.
Card 10 - Proportionality and Quantities
- Given a = 1/2 b and b = 1/3 c, and d = 3c, comparison yields equality between (a + b)/c and (a + b + c)/d.
Card 11 - Expression Analysis
- For a ≠ b, comparison of (a - b)/(b - a) results in a negative value.
- Therefore, Quantity B (which is 1) is greater.
Card 12 - Inequality and Expressions
- If x > y and x ≠ y, the comparison of ratios of squares indicates uncertainty in relative sizes.
- Further examination of the solution is recommended.
Card 13 - Inverse Proportionality Example
- If a is inversely proportional to b, then when a = 16 (b = 1), decreasing a to 8 results in b = 2.
Card 14 - Direct Proportionality Example
- Directly proportional case gives a = 8 when b = 2, thus for b = 4, a must equal 16.
Card 15 - Reiterating Inverse Proportionality
- Again considers time and students for erecting a bonfire; calculation confirms the time taken as 51 minutes.
Card 16 - Integer Sum Relationships
- Similar to Card 9, evaluates sum conditions with a + b = 75% of c, emphasizing potential for negative values complicating comparisons.
Card 17 - System of Equations
- From the equations a + b = 8, b + c = 11, and a + c = 5, summing all yields a + b + c = 12 as the solution.
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Test your knowledge with these tricky algebra flashcards from the Manhatten 5lb Book. Each card presents a challenging question designed to deepen your understanding of algebraic concepts. Challenge yourself and see how well you can solve these problems!
